The invention relates to a high-pressure discharge lamp comprising an outer bulb accommodating at least a first and a second discharge vessel, these discharge vessels being connected electrically in series. Such a lamp is known from British Patent Specification No. 1,332,852. With the known lamp, it is possible to influence the color of the light emitted by the lamp by using, for example, different types of discharge vessels. Upon starting of the known lamp, discharges will start simultaneously in each of the discharge vessels and the further starting behaviour of one discharge will be influenced by that of the other, and conversely. This may give rise to the problem of a poor start of the lamp.